Theorem. Nested blocks diagonalize the hierarchical precision [fdip-0099]

✍️sourceAGENTDRAFTED

On the path with vertices \(0,1,2,3,4,5\), let \(\mu >0\) and let the precision be the path Laplacian plus \(\mu \) on the diagonal. The diagonal entries are \(\mu +1\) at the two ends and \(\mu +2\) at the 4 interior vertices, and each of the 5 edges contributes \(-1\). Write \(m_0,m_1,m_2\) for the averages of the three adjacent pairs and \(d_0,d_1,d_2\) for half the differences, so that the field on pair \(i\) is \(m_i+d_i\) and \(m_i-d_i\). The change from these coordinates, means before differences, to the path coordinates has constant Jacobian \(8\). In these coordinates the quadratic form has blocks \[\begin {aligned} H_{mm}&=\begin {pmatrix}2\mu +1&-1&0\\-1&2\mu +2&-1\\0&-1&2\mu +1\end {pmatrix},\\ B&=\begin {pmatrix}-1&-1&0\\1&0&-1\\0&1&1\end {pmatrix},\\ D&=\begin {pmatrix}2\mu +5&1&0\\1&2\mu +6&1\\0&1&2\mu +5\end {pmatrix}, \end {aligned}\] where \(B\) holds the cross terms between means and differences. The marginal precision of the three means is the Schur complement \(H_{mm}-BD^{-1}B^{T}\).

The corner of \(H_{mm}\) vanishes. The corner of \(BD^{-1}B^{T}\) is the pairing of \((-1,-1,0)\) with the solution of \(D(x,y,z)^{T}=(0,1,1)^{T}\). The first row gives \(y=-(2\mu +5)x\). The last row then gives \(z=x+1/(2\mu +5)\). Substitution in the middle row produces \[-2(\mu +2)(2\mu +7)x=\frac {2(\mu +2)}{2\mu +5},\] hence \(x=-1/((2\mu +5)(2\mu +7))\) and \(y=1/(2\mu +7)\). The pairing equals the corner of \(BD^{-1}B^{T}\): \[-x-y=-\frac {2(\mu +2)}{(2\mu +5)(2\mu +7)}.\] Subtracting it from the zero corner of \(H_{mm}\) gives the positive quantity \[\frac {2(\mu +2)}{(2\mu +5)(2\mu +7)}.\] A nearest-neighbour precision on the three coarse vertices would have a zero corner. These pair means do not.

Let \(p\ge 2\) and \(k\ge 1\) be integers, and let the sites be the residues modulo \(p^k\). For \(0\le j\le k\) let \(E_j\) replace the value at each site by the mean of the sites congruent to it modulo \(p^{k-j}\). Each such class has \(p^j\) sites. Thus \(E_0\) is the identity, since congruence modulo \(p^k\) isolates every residue, and \(E_k\) replaces every value by the mean of all the sites. Each \(E_j\) is symmetric, because its matrix is constant on the pairs inside each class and zero across classes, and \(E_j^2=E_j\), because a class mean is unchanged by a second average. If \(i\ge j\), congruence modulo \(p^{k-i}\) is coarser than congruence modulo \(p^{k-j}\), so each coarser class is a union of finer classes. A function constant on a coarser class is constant on each finer class inside it, and therefore \(E_jE_i=E_i\). That identity does not use the sizes of the classes. For the other order, every finer class has \(p^j\) sites, so the mean on a coarser class is the mean of the finer means and \(E_iE_j=E_i\).

In particular \(E_{j+1}E_j=E_jE_{j+1}=E_{j+1}\). Set \(P_j=E_j-E_{j+1}\) for \(j<k\), and \(P_k=E_k\). Then \(P_jE_{j+1}=E_jE_{j+1}-E_{j+1}^2=E_{j+1}-E_{j+1}=0\). If \(i>j\), then \(i\ge j+1\). The size-free identity gives \(E_{j+1}E_i=E_i\), so \(E_if=E_{j+1}(E_if)\) and the image of \(E_i\) lies in the image of \(E_{j+1}\). When \(i<k\) the same left factor gives \(E_{j+1}E_{i+1}=E_{i+1}\), and both summands of \(P_i\) land in that image. When \(i=k\), \(E_{j+1}E_k=E_k\). A vector in the image of \(P_i\) is therefore \(E_{j+1}\) applied to something, and \(P_jE_{j+1}=0\) gives \(P_jP_i=0\). Both matrices are symmetric, so the opposite product vanishes as well. The case \(P_k=E_k\) is symmetric and idempotent because \(E_k\) is. For \(j<k\), the square expands to \(E_j-E_jE_{j+1}-E_{j+1}E_j+E_{j+1}^2\), and the relations replace that by \(E_j-E_{j+1}-E_{j+1}+E_{j+1}=P_j\). Symmetry of \(P_j\) is the same property of the two averages. The sum of the projectors telescopes to \(E_0\), since \(E_0\) is the identity. These matrices are orthogonal projectors, they are pairwise orthogonal, and they sum to the identity. For real coefficients \(c_0,\ldots ,c_k\), the hierarchical precision \[Q=\sum _{j=0}^{k}c_jP_j\] acts on the image of \(P_j\) by multiplication by \(c_j\). Whenever every coefficient is nonzero, \(Q\) is invertible and its inverse multiplies that same image by \(c_j^{-1}\).